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A ring in graph theory

Published online by Cambridge University Press:  24 October 2008

W. T. Tutte
W. T. Tutte
Affiliation:
Trinity CollegeCambridge
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Extract

We call a point set in a complex K a 0-cell if it contains just one point of K, and a 1-cell if it is an open arc. A set L of 0-cells and 1-cells of K is called a linear graph on K if

(i) no two members of L intersect,

(ii) the union of all the members of L is K,

(iii) each end-point of a 1-cell of L is a 0-cell of L

and (iv) the number of 0-cells and 1-cells of L is finite and not 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 1 , January 1947 , pp. 26 - 40
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

* The p 1(L) of this paper is Whitney's ‘nullity’, and p 0(L) is Whitney's P. The ‘components’ in this paper are Whitney's ‘pieces’: he uses the word ‘component’ with a different meaning. A footnote to Whitney's paper, dealing with some work of R. M. Forster, is particularly interesting with respect to the subject of the present paper.

* Whitney, Hassler, ‘A logical expansion in mathematics’, Bull. American Math. Soc. 38 (1932), 572–9.CrossRefGoogle Scholar

* See Lefschetz, , Algebraic Topology (Amer. Math. Soc. Colloquium Publications, vol. 27), p. 106.Google Scholar

† It may be mentioned that for graphs on the sphere a β-colouring is essentially equivalent to a colouring of the regions of the map defined by a graph in λ colours. The colours can be represented by elements of G and so the colouring can be represented by a 2-chain on the map with coefficients in G. A β-colouring is simply the boundary of such a 2-chain. The number of 1-cells incident with two regions of the same colour (or incident with only one region) in a given colouring is given by the number ψ(g G) where g G is the corresponding β-colouring.

* König, Dénes, Theorie der Endlichen und unendlichen Graphen (Leipzig, 1936), p. 186.Google Scholar